
\newenvironment{carlist}
 {\begin{list}{$\bullet$}
 {\setlength{\topsep}{0in} \setlength{\partopsep}{0in}
  \setlength{\parsep}{0in} \setlength{\itemsep}{\parskip}
  \setlength{\leftmargin}{0.07in} \setlength{\rightmargin}{0.08in}
  \setlength{\listparindent}{0in} \setlength{\labelwidth}{0.08in}
  \setlength{\labelsep}{0.1in} \setlength{\itemindent}{0in}}}
 {\end{list}}

\newcommand{\bcar}{\begin{carlist}}
\newcommand{\ecar}{\end{carlist}}

\newenvironment{carliste}
 {\begin{list}x
 {\setlength{\topsep}{0in} \setlength{\partopsep}{0in}
  \setlength{\parsep}{0in} \setlength{\itemsep}{\parskip}
  \setlength{\leftmargin}{0.07in} \setlength{\rightmargin}{0.08in}
  \setlength{\listparindent}{0in} \setlength{\labelwidth}{0.08in}
  \setlength{\labelsep}{0.1in} \setlength{\itemindent}{0in}}}
 {\end{list}}

\newcommand{\bcare}{\begin{carliste}}
\newcommand{\ecare}{\end{carliste}}


\newcommand{\mytit}[1]{\begin{center} \subsection*{#1} \end{center}}

\newcommand{\Prob}{\ensuremath{\mathbb{P}}}

% June 8, 2003


\newcommand{\Dualfun}{\ensuremath{\mathcal{Q}}}

\newcommand{\tracer}[2]{\ensuremath{\langle \!\langle {#1}, \; {#2}
\rangle \!\rangle}}

\newcommand{\Symcone}[1]{\ensuremath{\mathcal{S}^{#1}}}

% Vertex of polytope specified by configuration
\newcommand{\pvert}[2]{\ensuremath{#1_{#2}}}

% Base measure
\newcommand{\nub}{\ensuremath{{\boldsymbol{\nu}}}}
\newcommand{\Mdom}{\ensuremath{\mathcal{M}}}



% March 27, 2003

\newcommand{\mhalf}{\ensuremath{\frac{1}{2}}}
\newcommand{\ext}{\ensuremath{\operatorname{ext}}}
\newcommand{\co}{\ensuremath{\operatorname{co}}}


\newcommand{\Genapp}{\ensuremath{C}}
\newcommand{\Specapp}{\ensuremath{\Dualc_{app}}}
\newcommand{\Partapp}{\ensuremath{B}}
\newcommand{\Dualapp}{\ensuremath{\Partapp^*}}
\newcommand{\Appset}{\ensuremath{APP}}

% Basic command for clique
\newcommand{\clset}{\ensuremath{C}}

\newcommand{\aseta}{\ensuremath{A}}
\newcommand{\bseta}{\ensuremath{B}}
\newcommand{\cseta}{\ensuremath{C}}

\newcommand{\epvec}{\ensuremath{{\boldsymbol{\eparam}}}}
\newcommand{\epvecs}{\ensuremath{{\boldsymbol{\eparam}}^*}}

\newcommand{\myproj}[2]{\ensuremath{{#1}({#2})}}

\newcommand{\treedist}{\ensuremath{\rho}}
\newcommand{\treedistvec}{\ensuremath{{\boldsymbol{\rho}}}}
\newcommand{\trdistvec}{\ensuremath{{\boldsymbol{\rho_e}}}}

\newcommand{\grmaxmarg}{\ensuremath{\nu}}
\newcommand{\grmaxmargb}{\ensuremath{\nu}}

\newcommand{\support}[2]{\ensuremath{\delta^*(#1 \, | \, #2)}}
\newcommand{\indicate}[2]{\ensuremath{\delta(#1 \, | \, #2)}}

\newcommand{\supportsolo}{\ensuremath{\delta^*}}
\newcommand{\indicatesolo}{\ensuremath{\delta}}


\newcommand{\xp}{\ensuremath{x}}
\newcommand{\xd}{\ensuremath{x^*}}


\newcommand{\yp}{\ensuremath{y}}
\newcommand{\yd}{\ensuremath{y^*}}

\newcommand{\func}{\ensuremath{f}}
\newcommand{\funcdual}{\ensuremath{f^*}}





\newcommand{\realn}{\ensuremath{\real^n}}
\newcommand{\realnpos}{\ensuremath{\real_+^n}}
\newcommand{\realnspos}{\ensuremath{\real_{++}^n}}


\newcommand{\realpos}{\ensuremath{\real_+}}
\newcommand{\realspos}{\ensuremath{\real_{++}}}

\newcommand{\convset}{\ensuremath{C}}
\newcommand{\aff}{\ensuremath{\operatorname{aff}}}

\newcommand{\lama}{\ensuremath{\lambda}}
\newcommand{\lamb}{\ensuremath{\mu}}


\newcommand{\cone}{\ensuremath{K}}
\newcommand{\dcone}{\ensuremath{K^*}}

\newcommand{\xset}{\ensuremath{X}}

\newcommand{\con}{\ensuremath{c}}

\newcommand{\coneleq}{\ensuremath{\preceq_{\cone}}}
\newcommand{\conele}{\ensuremath{\prec_{\cone}}}
\newcommand{\conegeq}{\ensuremath{\succeq_{\cone}}}
\newcommand{\conege}{\ensuremath{\succ_{\cone}}}

\newcommand{\gauge}{\ensuremath{\gamma}}


% Measure
\newcommand{\meas}{\ensuremath{m}}
\newcommand{\power}{\ensuremath{\mathcal{P}}}
\newcommand{\measset}{\mathcal{M}}
\newcommand{\algeb}{\mathcal{A}}
\newcommand{\balgeb}{\mathcal{B}}
\newcommand{\measfam}{\mathcal{F}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% March 10, 2003

\newcommand{\op}[1]{\ensuremath{\operatorname{#1}}}

\newcommand{\Semidef}{\ensuremath{\operatorname{SDEF}}}

\newcommand{\Mat}{\ensuremath{M}}

\newcommand{\spnodenum}{\ensuremath{n}}

% Domain of function
\newcommand{\dom}{\ensuremath{\operatorname{dom}}}

% Moment mapping
\newcommand{\Mommap}{\ensuremath{\Lambda}}


\newcommand{\stsp}{\ensuremath{\statesp^{\spnodenum}}}


\newcommand{\Locone}{\ensuremath{\operatorname{LOCAL}}}
\newcommand{\Locset}{\ensuremath{\operatorname{LOCAL}}}

\newcommand{\treegr}{\ensuremath{T}}


\newcommand{\tausym}{\ensuremath{\tau}}
\newcommand{\tausymb}{\ensuremath{\tau}}

\newcommand{\specexp}{\ensuremath{\gamma^*}}
\newcommand{\logmaxspec}{\ensuremath{\ell^*}}

\newcommand{\mmax}{\operatorname{maximize}}
\newcommand{\mmin}{\operatorname{minimize}}
\newcommand{\subto}{\operatorname{subject} \; \operatorname{to}}

\newcommand{\Partcapp}{\ensuremath{B}}
\newcommand{\Dualcapp}{\ensuremath{\Partcapp^*}}

\newcommand{\Partctil}{\ensuremath{\wtil{\Partc}}}
\newcommand{\Dualctil}{\ensuremath{\wtil{\Partc}^*}}

\newcommand{\meanpar}{\ensuremath{\mu}}
\newcommand{\meanparb}{\ensuremath{\mu}}
\newcommand{\meanpartil}{\ensuremath{\wtil{\meanpar}}}
\newcommand{\meanparmf}{\ensuremath{\meanpar^{\operatorname{MF}}}}



% New commands as of February 11, 2003

% LP norms
\newcommand{\lp}[2]{\ensuremath{\|#2\|_{#1}}}

% Wedge and vee min and max
\newcommand{\wmin}{\ensuremath{\wedge}}
\newcommand{\vmax}{\ensuremath{\vee}}

\newcommand{\Outer}{\ensuremath{\operatorname{OUT}}}
\newcommand{\taupar}{\ensuremath{\tau}}



%%% New version of \caption puts things in smaller type, single-spaced 
%%% and indents them to set them off more from the text.
\makeatletter
\long\def\@makecaption#1#2{
        \vskip 0.8ex
        \setbox\@tempboxa\hbox{\small {\bf #1:} #2}
        \parindent 1.5em  %% How can we use the global value of this???
        \dimen0=\hsize
        \advance\dimen0 by -3em
        \ifdim \wd\@tempboxa >\dimen0
                \hbox to \hsize{
                        \parindent 0em
                        \hfil 
                        \parbox{\dimen0}{\def\baselinestretch{0.96}\small
                                {\bf #1.} #2
                                %%\unhbox\@tempboxa
                                } 
                        \hfil}
        \else \hbox to \hsize{\hfil \box\@tempboxa \hfil}
        \fi
        }
\makeatother



\long\def\comment#1{}

%% modify cite command to do \cite[e.g, ]{foo1,foo2} 
\makeatletter
\def\@cite#1#2{[\if@tempswa #2 \fi #1]}
\makeatother


\makeatletter
\long\def\barenote#1{
    \insert\footins{\footnotesize
    \interlinepenalty\interfootnotelinepenalty 
    \splittopskip\footnotesep
    \splitmaxdepth \dp\strutbox \floatingpenalty \@MM
    \hsize\columnwidth \@parboxrestore
    {\rule{\z@}{\footnotesep}\ignorespaces
              % indent
      #1\strut}}}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% BOOK CHAPTER


\newcommand{\Tractmarg}{\ensuremath{\operatorname{TRACT}}}
\newcommand{\Tractmargfact}{\ensuremath{\operatorname{FACT}}}
\newcommand{\mysubmani}{\ensuremath{{\mathcal{M}}_e}}

\newcommand{\subgraphfact}{\ensuremath{\subgraph_{0}}}


\newcommand{\Partdual}{\ensuremath{\Part^*}}

\newcommand{\Partc}{\ensuremath{A}}
\newcommand{\Dualc}{\ensuremath{\Partc^*}}

% Relative interior of a set
\newcommand{\relint}{\ensuremath{\operatorname{ri}}}

\newcommand{\interior}{\ensuremath{\operatorname{int}}}
\newcommand{\intr}{\ensuremath{\operatorname{int}}}

% Script I for indicator function
\newcommand{\Ind}{\ensuremath{\mathbb{I\,}}}

% Capitol \Mu for domain of mean parameter
\newcommand{\Mu}{\ensuremath{\mathcal{M}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\Hmargset}{\ensuremath{\operatorname{HYPER}}}


\newcommand{\bmatpos}{\ensuremath{\begin{bmatrix} 0.4 & 0.1 \\ 0.1 & 0.4 \end{bmatrix}}}


\newcommand{\bmatneg}{\ensuremath{\begin{bmatrix} 0.1 & 0.4 \\ 0.4 & 0.1 \end{bmatrix}}}

\newcommand{\bmatvec}{\ensuremath{\begin{bmatrix} 0.5 \\ 0.5 \end{bmatrix}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\newcommand{\Tempfun}{\ensuremath{{\mathcal{G}}}}

% New command: domain of the dual function
\newcommand{\dualdom}{\ensuremath{{\mathcal{D}}}}

\newcommand{\eparambp}{\ensuremath{\eparam^{BP}}}
\newcommand{\eparamml}{\ensuremath{\eparam^{ML}}}
\newcommand{\eparamre}{\ensuremath{\eparam^{RE}}}

% Number of observations
\newcommand{\obsnum}{\ensuremath{n}}
\newcommand{\girth}{\ensuremath{g}}
\newcommand{\girfun}{\ensuremath{l}}


\newcommand{\Loglike}{\ensuremath{L}}
\newcommand{\Apploglike}{\ensuremath{\wtil{\Loglike}}}



\newcommand{\cona}{\ensuremath{{\mathcal{C}_e}}}
\newcommand{\conb}{\ensuremath{{\mathcal{C}_m}}}


\newcommand{\modcom}{\ensuremath{\varphi}}
\newcommand{\modmescom}{\ensuremath{\rho}}

\newcommand{\const}{\ensuremath{C}}
\newcommand{\eparamest}{\ensuremath{\estim{\eparam}}}
\newcommand{\veparamest}{\ensuremath{\estim{\veparam}}}

\newcommand{\Corset}{\ensuremath{\operatorname{COR}}}

\newcommand{\trwmax}{tree-reweighted max-sum$\;$}
\newcommand{\trwmaxs}{tree-reweighted max-sum}

\newcommand{\trwmaxpr}{tree-reweighted max-product$\;$}
\newcommand{\trwmaxprs}{tree-reweighted max-product}



\newcommand{\examaxmarg}{\ensuremath{\wtil{\exafun}}}

\newcommand{\trmaxmarg}{\ensuremath{T}}
\newcommand{\trmaxmargb}{\ensuremath{\mathbf{\trmaxmarg}}}

\newcommand{\newtmarg}{\ensuremath{Q}}
\newcommand{\newtmargb}{\ensuremath{\mathbf{\newtmarg}}}

\newcommand{\trmaxm}{\ensuremath{\trmaxmarg^*}}
\newcommand{\trmaxmb}{\ensuremath{\trmaxmargb^*}}

\newcommand{\Dualfunc}{\ensuremath{{\mathcal{H}}}}



\newcommand{\Margset}{\ensuremath{\operatorname{MARG}}}
\newcommand{\Tmargset}{\ensuremath{\operatorname{TREE}}}
\newcommand{\infbit}{\ensuremath{v}}
\newcommand{\outbit}{\ensuremath{u}}
\newcommand{\infvec}{\ensuremath{\vvec}}
\newcommand{\outvec}{\ensuremath{\uvec}}
\newcommand{\cbit}{\ensuremath{C}}
\newcommand{\cbitvec}{\ensuremath{\mathbf{\cbit}}}
\newcommand{\cross}{\ensuremath{p}}


\newcommand{\xoptset}{\ensuremath{\operatorname{OPT}}}


\newcommand{\asfig}{0.1}
\newcommand{\bsfig}{0.9}

\newcommand{\csfig}{0.1}
\newcommand{\dsfig}{0.9}

\newcommand{\esfig}{0.4}
\newcommand{\fsfig}{0.6}

\newcommand{\eps}{\ensuremath{\epsilon}}


\newcommand{\bit}{\begin{itemize}}
\newcommand{\eit}{\end{itemize}}
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}

\newcommand{\bear}{\begin{eqnarray}}
\newcommand{\eear}{\end{eqnarray}}

\newcommand{\htiny}{\hspace*{.1in}}
\newcommand{\vtiny}{\vspace*{.1in}}
\newcommand{\vsmall}{\vspace*{.2in}}
\newcommand{\hsmall}{\hspace*{.2in}}

\newcommand{\fn}{\footnotesize}
%\newcommand{\myspec}[2]{\ensuremath{\frac{\treefun_{#1 #2}}{\treefun_{#1} \, \treefun_{#2}}}}
%\newcommand{\myspec}[2]{\ensuremath{\compat_{#1 #2}}}


\newcommand{\myspec}[2]{\ensuremath{\frac{\treefun^*_{#1 #2}}{\treefun^*_{#1} \, \treefun^*_{#2}}}}
\newcommand{\mysing}[1]{\ensuremath{\treefun^*_{#1}}}


\newcommand{\myspectwo}[2]{\ensuremath{\compat_{#1 #2}}}


%%%%%%%%%%%%%%%%%
% Macros defined for this paper
\newcommand{\xhatcovar}{\ensuremath{\estim{P}}}
\newcommand{\bigB}{\ensuremath{{\bf{B}}}}
\newcommand{\bigF}{\ensuremath{{\bf{F}}}}
\newcommand{\alglong}{embedded trees }
\newcommand{\algshortnips}{ET }
\newcommand{\algshortnip}{ET}
\newcommand{\len}{\ensuremath{l}}
\newcommand{\lenx}{\ensuremath{L_\xvec}}
\newcommand{\errorvec}{\ensuremath{\wtil{\evec}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\scaler}{\ensuremath{i}}
\newcommand{\maxscaler}{\ensuremath{I}}
\newcommand{\spinparam}{\ensuremath{\omega}}

%%%%%%%%%%%%%%%%%%% COMMANDS FROM TRP PAPER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\rterm}{\ensuremath{r}}

\newcommand{\bpmapbold}{\ensuremath{{H}}}
\newcommand{\bpmap}{\ensuremath{H}}
\newcommand{\bfzero}{\ensuremath{\mathbf{0}}}
\newcommand{\gaminit}{\ensuremath{\bar{\gamma}^0}}
\newcommand{\nodevar}{\ensuremath{r}}
\newcommand{\qform}{\ensuremath{A}} 
\newcommand{\qformm}{\ensuremath{a}} 
\newcommand{\pdf}{pdf$\;\,$}
\newcommand{\mycompat}{compatibility function$\;$}
\newcommand{\mycompats}{compatibility functions$\;$}
%\newcommand{\setp}{\ensuremath{{{\mathcal{P}}}}}

%\newcommand{\inject}{\ensuremath{\mathcal{I}}}
%\newcommand{\estraint}{\ensuremath{\mathcal{D}}}

\newcommand{\tiltpart}{\ensuremath{F}}
\newcommand{\gammabar}{\ensuremath{\bar{\gamma}}}
\newcommand{\sett}{\ensuremath{\mathcal{U}}}
\newcommand{\etabar}{\ensuremath{\bar{\eta}}}
\newcommand{\etabarbf}{\ensuremath{\mathbf{\bar{\eta}}}}
%\newcommand{\Dualpart}{\ensuremath{\Psi}}

\newcommand{\Hacmap}{\ensuremath{\underline{\Lambda}}}
\newcommand{\underest}{\ensuremath{\underline{\estimate}}}

% Delta function for node x_s = j
\newcommand{\delf}{\ensuremath{\delta(x_s = j)}}
\newcommand{\dellf}[2]{\ensuremath{\delta(x_{#1}={#2})}}


\newcommand{\algsing}{reparameterization }
\newcommand{\Algsing}{Reparameterization }


\newcommand{\hackint}{\ensuremath{K}}




%%%%


\newcommand{\tilt}[2]{\ensuremath{{#1}_{#2}}}
\newcommand{\Partpl}[1]{\ensuremath{\Part_{#1}}}

\newcommand{\proofbox}{\begin{flushright} $\Box$ \end{flushright}}
\newcommand{\sptreenum}{\ensuremath{\mathcal{S}}}
\newcommand{\numcut}{\ensuremath{c}}

\newcommand{\partnum}{\ensuremath{L}}

\newcommand{\aind}{\ensuremath{a}}
\newcommand{\bind}{\ensuremath{b}}
\newcommand{\cind}{\ensuremath{c}}

\newcommand{\uni}{\ensuremath{\mathcal{U}}}
\newcommand{\edstrength}{\ensuremath{S}}

\newcommand{\supp}{\ensuremath{\operatorname{supp}}}


\newcommand{\evec}{\ensuremath{{\mathbf{e}}}}
\newcommand{\tvec}{\ensuremath{{\mathbf{t}}}}
\newcommand{\avec}{\ensuremath{{{\mathbf{a}}}}}
\newcommand{\bvec}{\ensuremath{{{\mathbf{b}}}}}
\newcommand{\cvec}{\ensuremath{{{\mathbf{c}}}}}
\newcommand{\clipotvec}{\ensuremath{{\boldsymbol{\clipot}}}}
\newcommand{\compatvec}{\ensuremath{{\boldsymbol{\compat}}}}

\newcommand{\Tiltpart}{\ensuremath{\Part}}
\newcommand{\simplex}{\ensuremath{\mathcal{P}}}

\newcommand{\Info}{\ensuremath{I}}
\newcommand{\Ent}{\ensuremath{H}}

\newcommand{\upoptfun}{\ensuremath{\mathcal{F}}}
\newcommand{\Mufun}{\ensuremath{\mathcal{H}}}
\newcommand{\treepoly}{\ensuremath{\mathbb{T}}}
\newcommand{\forestpoly}{\ensuremath{\mathbb{F}}}

\newcommand{\treepolypos}{\ensuremath{\mathbb{T}_+}}
\newcommand{\Mfun}{\ensuremath{\mathcal{L}}}
\newcommand{\Finfun}{\ensuremath{\mathcal{F}}}

\newcommand{\treewei}{\ensuremath{\mathbb{T}}}

\newcommand{\treewidth}{\ensuremath{t}}

\newcommand{\meanpoly}{\ensuremath{\mathbb{L}}}
\newcommand{\truepoly}{\ensuremath{\mathbb{M}}}


\newcommand{\maxset}{\ensuremath{\mathbb{M}}}
\newcommand{\optset}{\ensuremath{\operatorname{OPT}}}
\newcommand{\conv}{\ensuremath{\operatorname{conv}}}

\newcommand{\Tfunc}{\ensuremath{F}}
\newcommand{\Jfunc}{\ensuremath{J}}

\newcommand{\edind}{\ensuremath{e}}

\newcommand{\hypergraph}{\ensuremath{G_{\operatorname{HYP}}}}
\newcommand{\hypergraphcal}{\ensuremath{\mathcal{H}}}
\newcommand{\hyperedge}{\ensuremath{h}}
\newcommand{\hyp}{\ensuremath{\edge}}
\newcommand{\hypertree}{\ensuremath{\mathcal{T}}}
\newcommand{\hyperedtwo}{\ensuremath{g}}
\newcommand{\hyperedth}{\ensuremath{f}}

\newcommand{\recur}{\ensuremath{\varphi}}


\newcommand{\apdist}{\ensuremath{{\mathcal{T}}}}
\newcommand{\tract}{\mathfrak{T}}
\newcommand{\dweivec}{\ensuremath{{\boldsymbol{{\vec{\dwei}}}}}}
%\newcommand{\dweivecfx}{\ensuremath{{\boldsymbol{\bar{\dwei}}}}}
\newcommand{\dweivecfx}{\ensuremath{\dweivec}}


\newcommand{\cpar}{\ensuremath{\beta}}

\newcommand{\dwvec}{\ensuremath{\boldsymbol{\dwei_e}}}
\newcommand{\dwvech}{\ensuremath{\boldsymbol{\dwei_\hyperedge}}}
%\newcommand{\dwvecfx}{\ensuremath{\boldsymbol{{\bar{\dwei}_e}}}}
\newcommand{\dwvecfx}{\ensuremath{\dwvec}}


\newcommand{\spinc}{\ensuremath{\boldsymbol{\nu}}}
\newcommand{\dualfun}{\ensuremath{{{\mathcal{Q}}}}}
\newcommand{\lagmul}{\ensuremath{\lambda}}
\newcommand{\lamvec}{\ensuremath{{\boldsymbol{\lambda}}}}
\newcommand{\lagtwo}{\ensuremath{\xi}}
\newcommand{\veparam}{\ensuremath{\Theta}}
\newcommand{\veparams}{\ensuremath{\veparam^*}}
%\newcommand{\dweifx}{\ensuremath{\bar{\dwei}}}
\newcommand{\dweifx}{\ensuremath{\dwei}}


\newcommand{\mubf}{\ensuremath{\boldsymbol{\mu}}}

\newcommand{\Lag}{\ensuremath{{\mathcal{L}}}}

\newcommand{\upb}{\ensuremath{\mathcal{U}}}

\newcommand{\qsumind}{\ensuremath{\Lambda}}

\newcommand{\undisa}{\ensuremath{\lambda}}
\newcommand{\undisb}{\ensuremath{\lambda}}

\newcommand{\paramcore}{\ensuremath{\eparam_{\coreset}}}
\newcommand{\parampl}[1]{\ensuremath{\eparam_{\coreset \; \cup \:
#1}}}

\newcommand{\messexp}{\ensuremath{\nu}}
\newcommand{\edvec}{\ensuremath{\Sigma}}

\newcommand{\hparamex}{\ensuremath{\bar{\eparam}}}

\newcommand{\orr}{\operatorname{or}}

\newcommand{\rsumind}{\ensuremath{\Upsilon}}
\newcommand{\Marg}{\ensuremath{\mathbb{{M}}}}
\newcommand{\pairpoly}{\ensuremath{\mathfrak{{P}}}}

\newcommand{\cliqueop}{{\operatorname{clique}}}
\newcommand{\sepop}{{\operatorname{separator}}}
\newcommand{\sepp}{{\operatorname{sep}}}

\newcommand{\core}{{\operatorname{core}}}
\newcommand{\full}{{\operatorname{full}}}
\newcommand{\qdist}{\ensuremath{Q}}
\newcommand{\qdisttil}{\ensuremath{\wtil{\qdist}}}
\newcommand{\qdistaux}{\ensuremath{\wtil{\qdist}}}

\newcommand{\qdistvec}{\ensuremath{\mathbf{\vec{\qdist}}}}
%\newcommand{\qdistvec}{\ensuremath{\mathbf{\qdist}}} 

\newcommand{\qdistvecs}{\ensuremath{\qdistvec^*}}

\newcommand{\qcore}{\ensuremath{\qdist_{\coreset}}}
\newcommand{\qcorepl}[1]{\ensuremath{\qdist_{\coreset \: \cup \; #1}}}


\newcommand{\edset}{\ensuremath{\Delta}}
\newcommand{\edsettr}{\ensuremath{\wtil{\edset}}}

\newcommand{\resid}{\ensuremath{\mathbf{R}}}
\newcommand{\residtr}{\ensuremath{\wtil{\mathbf{R}}}}
\newcommand{\coreset}{\ensuremath{\mathbf{A}}}
\newcommand{\genset}{\ensuremath{\mathbf{B}}}
\newcommand{\mybull}{\ensuremath{\bullet}}

\newcommand{\pcore}{\ensuremath{\pdist_{\coreset}}}
\newcommand{\pcorepl}[1]{\ensuremath{\pdist_{\coreset\: \cup \; #1}}}



\newcommand{\pcorelabel}{CTD}
\newcommand{\qcorelabel}{CAD}
\newcommand{\pcorelabels}{CTD$\;$}
\newcommand{\qcorelabels}{CAD$\;$}

\newcommand{\pdist}{\ensuremath{P}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% New stuff from August 9, 2001

\newcommand{\binn}{\ensuremath{\{0,1\}^\nodenum}}

\newcommand{\famdist}{\ensuremath{\mathcal{F}}}
\newcommand{\params}{{\ensuremath{\param^{*}}}}

\newcommand{\eflatman}{\ensuremath{\mathcal{M}_e}}
\newcommand{\mflatman}{\ensuremath{\mathcal{M}_m}}


\newcommand{\setp}{\ensuremath{{\mathcal{R}}}}


\newcommand{\Legmap}{\ensuremath{\Lambda}}


\newcommand{\modpart}{\ensuremath{F}}
\newcommand{\Npart}{\ensuremath{\mathcal{F}}}

\newcommand{\Partinf}{\ensuremath{\Part_{\infty}}}

% New stuff from December 28, 2000

\newcommand{\df}{\ensuremath{d}}

\newcommand{\alphnorm}{\ensuremath{\kappa}}
\newcommand{\mynorm}[1]{\ensuremath{\alphnorm_{#1}}}

%\newcommand{\bethe}{\ensuremath{G}}

\newcommand{\bethe}{\ensuremath{{\mathcal{G}}}}
\newcommand{\gfunc}{\ensuremath{{\mathcal{G}}}}

\newcommand{\mybethe}{\ensuremath{{\mathcal{G}}}}
\newcommand{\bigbethe}[2]{\ensuremath{\mybethe_{{#1}; \, {#2}}}}
\newcommand{\bigbethenon}[2]{\ensuremath{\wtil{\mybethe}_{{#1}; \, {#2}}}}

\newcommand{\step}{\ensuremath{\lambda}}
\newcommand{\straint}{\ensuremath{{\mathbb{C}}}}
\newcommand{\estraint}{\ensuremath{{\mathbb{D}}}}
\newcommand{\kullind}[3]{\ensuremath{D^{#1}({#2}\; \| {#3})}}

\newcommand{\fullcompose}[1]{\ensuremath{(\estimate \circ\project)^{#1}}}
\newcommand{\compose}{\ensuremath{{\mathcal{Q}}}}

\newcommand{\eparams}{{\ensuremath{\eparam^*}}}
\newcommand{\meparams}{{\ensuremath{\meparam^*}}}
\newcommand{\meparamtil}{{\ensuremath{{\wtil{\meparam}}}}}

\newcommand{\vecgamma}{{\ensuremath{\boldsymbol{\gamma}}}}
\newcommand{\vecgammatil}{{\ensuremath{\boldsymbol{\gammatil}}}}

\newcommand{\vgammainf}{{\ensuremath{\gamma}}}
\newcommand{\vgammainftil}{{\ensuremath{\gammatil}}}


\newcommand{\jind}{\ensuremath{k}}

\newcommand{\Thetmap}{\ensuremath{\Theta}}
\newcommand{\Pmap}{\ensuremath{\Legmap}}

\newcommand{\breg}[3]{\ensuremath{B_{#1}(#2; \; #3)}}
\newcommand{\bregsin}[1]{\ensuremath{B_{#1}}}


\newcommand{\algnames}{tree-based reparameterization$\,$}
\newcommand{\algname}{tree-based reparameterization}
\newcommand{\algshorts}{TRP$\;$} 
\newcommand{\algshort}{TRP}

\newcommand{\gmps}{GMP$\;$} 
\newcommand{\gmp}{GMP}


% New stuff as of Dec 9, 2000; Version d_merge.tex
\newcommand{\itind}{\ensuremath{i}}
\newcommand{\mytind}{\ensuremath{i}}
\newcommand{\spind}{\ensuremath{l}}

\newcommand{\opteparam}{\ensuremath{\eparam^\dagger}}
\newcommand{\opttreefun}{\ensuremath{\treefun^\dagger}}
\newcommand{\eparambar}{\ensuremath{\bar{\eparam}}}
\newcommand{\eparamtil}{\ensuremath{{\wtil{\eparam}}}}
\newcommand{\gammatil}{\ensuremath{\wtil{\gamma}}}


\newcommand{\sumind}{\ensuremath{\alpha}}


\newcommand{\apmarg}[2]{\ensuremath{T^*_{#1; #2}}}
\newcommand{\trmarg}[2]{\ensuremath{P_{#1; #2}}}

\newcommand{\funa}{\ensuremath{\Lambda}}
\newcommand{\funb}{\ensuremath{\Psi}}

\newcommand{\etreefun}{\ensuremath{\estim{\treefun}}}
\newcommand{\treefuns}{\ensuremath{\treefun^*}}
\newcommand{\family}{\ensuremath{\scr{F}}}

\newcommand{\mymani}{\ensuremath{{\scr{M}}}}

\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\is}{is}
\DeclareMathOperator{\markov}{Markov}
\DeclareMathOperator{\wrt}{w.r.t}
\DeclareMathOperator{\suchthat}{s.t.}


\newcommand{\moeb}{M\"{o}bius $\,$}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%









%%%%%%%%%%
\newcommand{\bigM}{\ensuremath{{\bf{A}}}}


\newcommand{\xfix}{\ensuremath{\xsca^F}}
\newcommand{\xsam}{\ensuremath{\xsca^S}}
\newcommand{\yvecplus}{\ensuremath{\ysca^+}}

\newcommand{\myerr}{\ensuremath{E}}

\newcommand{\myinferr}{\ensuremath{\operatorname{Err}}}


\newcommand{\myinfdiff}{\ensuremath{\operatorname{Diff}}}


\newcommand{\desc}{\mathbf{d}}

\newcommand{\cliquenum}{\ensuremath{S}}
\newcommand{\statenum}{{\ensuremath{m}}}
\newcommand{\nodenum}{\ensuremath{N}}
\newcommand{\edgenum}{\ensuremath{|\edge|}}
\newcommand{\treenum}{\ensuremath{L}}

\newcommand{\tind}{\ensuremath{i}}
\newcommand{\tindset}{\ensuremath{{\mathcal{I}}}}

\newcommand{\boumat}{\ensuremath{{\mathcal{M}}}}

\newcommand{\coeff}[4]{\ensuremath{C\big(#1, \,#2; \; #3, \,#4\big)}}
\newcommand{\bcoeff}[5]{\ensuremath{D\big (#1, \,#2; \; #3, \,#4; \, #5\big)}}

\newcommand{\usetree}[2]{\ensuremath{\estim{#1}^{#2}}}
\newcommand{\usecut}[3]{\ensuremath{\wtil{#1}^{#2;#3}}}

\newcommand{\mess}{\ensuremath{M}}
\newcommand{\messvec}{\ensuremath{\mathbf{M}}}

\newcommand{\lmess}{\ensuremath{\zeta}}
\newcommand{\lmessvec}{\ensuremath{\boldsymbol{\zeta}}}

\newcommand{\linf}[1]{\ensuremath{\|{#1}\|_\infty}}

\newcommand{\messb}{\ensuremath{\mathbf{M}}}
%\newcommand{\neigh}{\ensuremath{\mathcal{N}}}
\newcommand{\neigh}{\ensuremath{\Gamma}}

\newcommand{\cneigh}{\ensuremath{{\mathcal{A}}}}
\newcommand{\covset}{\ensuremath{{\mathcal{C}}}}
\newcommand{\compat}{\ensuremath{\psi}}
\newcommand{\clipot}{\ensuremath{\phi}}
\newcommand{\clipottil}{\ensuremath{\wtil{\clipot}}}
\newcommand{\obsfun}{\ensuremath{\varphi}}
\newcommand{\obsmod}{\ensuremath{\Delta}}
\newcommand{\prob}{\ensuremath{{\mathbb{P}}}}
\newcommand{\genprob}{\ensuremath{{\mathcal{P'}}}}

\newcommand{\dualvar}{\ensuremath{\eta}}
\newcommand{\dualvartil}{\ensuremath{{\wtil{\dualvar}}}}
\newcommand{\dualvarbar}{\ensuremath{\bar{\dualvar}}}

\newcommand{\dualvarbold}{\ensuremath{{\boldsymbol{\eta}}}}
\newcommand{\Dual}{\ensuremath{\Psi}}
\newcommand{\dualvartree}{\ensuremath{\dualvar^{\tree}}}


\newcommand{\spprob}{\ensuremath{\mathcal{P}}}
\newcommand{\mydistset}{\ensuremath{\mathcal{M}}}

\newcommand{\eparam}{\ensuremath{\theta}}
\newcommand{\Eparam}{\ensuremath{\Theta}}
\newcommand{\Dome}{\ensuremath{\Theta}}
%\newcommand{\meparam}{\ensuremath{{{\boldsymbol{\theta}}}}}
\newcommand{\meparam}{\ensuremath{\eparam}}

\newcommand{\param}{\ensuremath{\gamma}}
\newcommand{\objec}{\ensuremath{F}}
\newcommand{\oparg}{\ensuremath{z}}
\newcommand{\approxnum}{\ensuremath{T}}
\newcommand{\taypoint}{\ensuremath{\bar{\nu}}}

\newcommand{\mani}{\ensuremath{{\scr{M}}}}
\newcommand{\statesp}{\ensuremath{{\scr{X}}}}
\newcommand{\ystatesp}{\ensuremath{{\scr{Y}}}}
\newcommand{\qstatesp}{\ensuremath{{\scr{\estim{X}}}}}

\newcommand{\statespn}{\ensuremath{\statesp^{\nodenum}}}
\newcommand{\ystatespn}{\ensuremath{\ystatesp^{\nodenum}}}

\newcommand{\qstatespn}{\ensuremath{\qstatesp^{\nodenum}}}

\newcommand{\tang}{\ensuremath{\scr{T}}}
\newcommand{\natbas}{\ensuremath{e}}
\newcommand{\Fisher}{\ensuremath{G}}
\newcommand{\fisher}{\ensuremath{g}}
\newcommand{\inter}{\ensuremath{\nu}}
\newcommand{\Part}{\ensuremath{\Phi}}
\newcommand{\Dualpart}{\ensuremath{\Psi}}


\newcommand{\iterfun}{\ensuremath{U}}
\newcommand{\biterfun}{\mathbf{U}}
\newcommand{\project}{\ensuremath{\Pi}}
\newcommand{\estimate}{{\mathcal{R}}}
\newcommand{\inject}{{\mathcal{I}}}

\newcommand{\optfun}{\ensuremath{P}}
\newcommand{\boptfun}{\mathbf{P}}

\newcommand{\iterfunb}{\ensuremath{\mathbf{U}}}
\newcommand{\biterfunb}{\mathbf{V}}

\newcommand{\iterinfun}{\ensuremath{{U}}}
\newcommand{\biterinfun}{{V}}


\newcommand{\sptree}{{\mathcal{T}}}
\newcommand{\caltree}{{\mathcal{T}}}
\newcommand{\cutset}{{\mathcal{C}}}
\newcommand{\super}{{\mathcal{F}}}

% Eigenvector / eigenvalue related notation
\newcommand{\eig}[1]{\ensuremath{\lambda_{#1}}}
\newcommand{\eigmax}{\ensuremath{\eig{\max}}}
\newcommand{\eigmin}{\ensuremath{\eig{\min}}}
\newcommand{\eigset}[1]{\ensuremath{\sigma\left({#1}\right)}}
\newcommand{\eigrad}[1]{\ensuremath{\rho\left({#1}\right)}}
\newcommand{\eigval}[1]{\ensuremath{\lambda_{#1}}}
\newcommand{\eigvalset}[1]{\ensuremath{\left\{\eigval{i}\left(#1\right)\right\}}}
\newcommand{\singval}[1]{\ensuremath{\sigma_{#1}}}
\newcommand{\singvalset}[1]{\ensuremath{\left\{\singval{i}\left(#1\right)\right\}}}
\newcommand{\bel}{\ensuremath{B}}

\newcommand{\treefun}{\ensuremath{T}}

%\newcommand{\treefunb}{\ensuremath{{\mathbf{T}}}}
\newcommand{\treefunb}{\ensuremath{T}}

\newcommand{\treeinfun}{\ensuremath{T}}


\newcommand{\exafun}{\ensuremath{P}}
%\newcommand{\exafunb}{\ensuremath{{{\bf{\exafun}}}}}
\newcommand{\exafunb}{\ensuremath{\exafun}}


\newcommand{\exainfun}{\ensuremath{\exafun}}


\newcommand{\indset}{\ensuremath{\scr{I}}}


\newcommand{\condprob}[3]{\ensuremath{p_{#1}(#2 | #3)}}

\newcommand{\sinprob}[2]{\ensuremath{p_{#1}(#2)}}

\newcommand{\pcond}[2]{\ensuremath{p(#1 \, | \, #2 )}}



\newcommand{\xcovar}{\ensuremath{P}}
\newcommand{\ncovar}{\ensuremath{R}}


\newcommand{\asetpl}{\ensuremath{A}}

\newcommand{\aset}{\ensuremath{{\scr{A}}}}
\newcommand{\bset}{\ensuremath{{\scr{B}}}}
\newcommand{\cset}{\ensuremath{{{\scr{C}}}}}

\newcommand{\aseto}{\ensuremath{A}}
\newcommand{\bseto}{\ensuremath{B}}
\newcommand{\cseto}{\ensuremath{C}}




\newcommand{\iset}{\ensuremath{\scr{I}}}
\newcommand{\feaset}{\ensuremath{\scr{A}}}

\newcommand{\jset}{\ensuremath{\scr{J}}}
\newcommand{\sset}{\ensuremath{\scr{S}}}

\newcommand{\csetaux}{\ensuremath{{\mathcal{D}}}}


%\newcommand{\cona}{\ensuremath{\mathcal{A}}}
%\newcommand{\conb}{\ensuremath{\mathcal{B}}}

% Little hacky command for the index of a weight element
\newcommand{\ind}{\ensuremath{i}}



% Little hacky command for the weight of a distribution
\newcommand{\dwei}{\ensuremath{\mu}}


\newcommand{\etalg}{ET }

% Symbol for the invariant distribution of a Markov chain
\newcommand{\ddist}{\pi}
\newcommand{\agree}{\ensuremath{\mathcal{A}}}

\newcommand{\order}{{\mathcal{O}}}
\newcommand{\bigs}{\phi}




\newcommand{\clique}{{\mathcal{C}}}
\newcommand{\cliqueset}{\mathbf{C}}


\newcommand{\sep}{{\mathcal{S}}}
\newcommand{\septwo}{{\mathcal{U}}}
\newcommand{\sepset}{{\mathbf{S}}}
\newcommand{\sepsettr}{\mathbf{\wtil{S}}}

\newcommand{\sepsetnew}{\ensuremath{\cliqueset_{\operatorname{sep}}}}

\newcommand{\Cut}[1]{\ensuremath{K_{#1}}}
\newcommand{\Kut}[1]{\ensuremath{K_{#1}}}
\newcommand{\Jesst}{\ensuremath{\Jestt}}
\newcommand{\Jestt}[1]{\ensuremath{\widehat{J}_{#1}}}
\newcommand{\Jestv}{\ensuremath{\widehat{J}_{\operatorname{orig}}}}
\newcommand{\estv}{\ensuremath{\estim{\xvec}}}
\newcommand{\xmap}{\ensuremath{\estv_{\operatorname{MAP}}}}
\newcommand{\xmaps}{\ensuremath{\estim{x}}}

% Matrices governing convergence of ET iterations
\newcommand{\convMatE}{\ensuremath{\mathbf{E}}}
\newcommand{\convMatD}{\ensuremath{\mathbf{D}}}
\newcommand{\convMatR}{\ensuremath{\mathbf{E}_{r}}}

\newcommand{\Fact}{\ensuremath{\Pi}}
%\newcommand{\tree}{\ensuremath{\scr{T}}}

\newcommand{\pather}{\ensuremath{\theta}}
\newcommand{\Path}{\ensuremath{\Theta}}

\newcommand{\ErrCov}{\ensuremath{\estim{P}_{full}}}


\newcommand{\inprod}[2]{\ensuremath{\langle #1 , \, #2 \rangle}}

\newcommand{\inprodd}[3]{\ensuremath{\langle #1 , \, #2 \rangle_{#3}}}

\newcommand{\covv}[3]{\ensuremath{\cov_{#3}\{#1, \, #2\}}}


\newcommand{\kull}[2]{\ensuremath{D(#1\; \| \; #2)}}

% IEEE Trans. Info. Theory form of Kullback-Leibler
\newcommand{\kullinf}[2]{\ensuremath{D(#1\, \| \, #2)}}


\newcommand{\normspec}[2]{\ensuremath{\|#1\|_{#2}}}


\newcommand{\joint}{{\cal A}}

\newcommand{\jsdata}{\ensuremath{{y}}}
\newcommand{\jsmean}{\ensuremath{{c}}}
\newcommand{\jsnoise}{\ensuremath{{v}}}

\newcommand{\wave}{\ensuremath{c}}
\newcommand{\wavevec}{\ensuremath{{\bf{c}}}}
\newcommand{\bigpro}{\ensuremath{\mathcal{I}}}

\newcommand{\ones}{\ensuremath{\mathbf{1}}}
\newcommand{\zeroes}{\ensuremath{\mathbf{0}}}
\newcommand{\laplacer}{\ensuremath{\mathcal{L}}}


\newcommand{\chooser}[2]{\ensuremath{\big(\frac{#1}{#2}\big)}}


% graph commands

%\newcommand{\subgraph}{\ensuremath{\mathcal{H}}}
%\newcommand{\vertex}{\ensuremath{\mathcal{V}}}
%\newcommand{\edge}{\ensuremath{\mathcal{E}}}
%\newcommand{\edgesub}{\ensuremath{{F}}}


%\newcommand{\graph}{\ensuremath{\mathcal{G}}}
\newcommand{\graph}{\ensuremath{G}}
\newcommand{\subgraph}{\ensuremath{H}}
\newcommand{\vertex}{\ensuremath{V}}
\newcommand{\edge}{\ensuremath{E}}
\newcommand{\edgesub}{\ensuremath{{F}}}


%\newcommand{\pathsym}{\ensuremath{\mathcal{P}}}
\newcommand{\pathsym}{\ensuremath{P}}

\newcommand{\graphtriang}{\ensuremath{\wtil{\graph}}}
\newcommand{\graphtr}{\ensuremath{\wtil{\graph}}}
\newcommand{\cliquesettr}{\ensuremath{\mathbf{\wtil{C}}}}

\newcommand{\ranker}{\ensuremath{r}}
\newcommand{\connect}{\ensuremath{c}}
\newcommand{\nuller}{\ensuremath{n}}
\newcommand{\numvertex}{\ensuremath{v}}




\newcommand{\treecal}{\ensuremath{\mathcal{T}}}


% Graph weighting matrix
\newcommand{\weight}{\ensuremath{\omega}}
\newcommand{\weights}{\ensuremath{w}}

\newcommand{\nobs}{\ensuremath{h}}

\newcommand{\normer}{\ensuremath{Z}}

\newcommand{\bk}{\ensuremath{\backslash}}
\newcommand{\kcon}{\ensuremath{K}}

%  Dyadic and Quad. Tree commands
\newcommand{\pare}{\ensuremath{\bar{\gamma}}}
\newcommand{\childa}[1]{\ensuremath{\alpha_{#1}}}
\newcommand{\ca}{\ensuremath{\bar{\alpha_1}}}
\newcommand{\cb}{\ensuremath{\bar{\alpha_2}}}
%\newcommand{\cc}{\ensuremath{\bar{\alpha_3}}}
\newcommand{\cd}{\ensuremath{\bar{\alpha_4}}}
\newcommand{\ancest}[2]{\ensuremath{{#1} \wedge {#2}}}
\newcommand{\childset}{\ensuremath{\operatorname{Ch}}}

\newcommand{\pars}{\ensuremath{\operatorname{pa}}}
\newcommand{\edif}{\ensuremath{\Delta}}

\newcommand{\parentt}{\ensuremath{\operatorname{parent}}}
\newcommand{\iss}{\ensuremath{\operatorname{is}}}
\newcommand{\of}{\ensuremath{\operatorname{of}}}

\newcommand{\future}[1]{\ensuremath{\yvec^{+}_{#1}}}
\newcommand{\past}[1]{\ensuremath{\yvec^{-}_{#1}}}
\newcommand{\lamplus}[2]{\ensuremath{\lambda^{+}_{#1}(#2({#1}))}}
\newcommand{\lamhatplus}[3]{\ensuremath{\widehat{\lambda}^{+}_{#1}(#2({#3}))}}
\newcommand{\aplus}[2]{\ensuremath{{#1}^{+}_{#2}}}
\newcommand{\aplushat}[2]{\ensuremath{\widehat{#1}^{+}_{#2}}}

\newcommand{\clamplus}{compplus}
\newcommand{\clamhat}{comphat}



\newcommand{\data}{\ensuremath{\yvec}}
\newcommand{\probd}[2]{\ensuremath{p(#1 \, | \, #2 )}}



\newcommand{\state}{\ensuremath{x}}
\newcommand{\obs}{\ensuremath{y}}



\newcommand{\inv}[1]{\ensuremath{#1^{-1}}}
\newcommand{\kry}{\ensuremath{\mathcal{K}}}

% Self similarity parameter
\newcommand{\selfsim}{\ensuremath{\lambda}}
% Tail parameter
\newcommand{\logexp}{\ensuremath{\beta}}

% Logs etc.
\newcommand{\logtwo}{\ensuremath{\log_2}}

\newcommand{\emax}{\lambda_{max}}
\newcommand{\emin}{\lambda_{min}}
\newcommand{\earb}[1]{\lambda_{#1}}

% Limits with stackrel
\newcommand{\convp}{\ensuremath{\stackrel}{\mathcal{P}}{\longrightarrow}}
\newcommand{\convd}{\ensuremath{\stackrel}{\mathitalic{d}}{\longrightarrow}}

\newcommand{\pderiv}[2]{\ensuremath{\frac{\partial{#1}}{\partial{#2}}}}
\newcommand{\pderivtwo}[3]{\ensuremath{\frac{\partial^2{#1}}{\partial{#2}
\, \partial{#3}}}}
\newcommand{\pderivtwod}[2]{\ensuremath{\frac{\partial^2{#1}}{\partial{#2}^2}}}

\newcommand{\pdif}[2]{\ensuremath{\frac{\partial{#1}}{\partial{#2}}}}
\newcommand{\pdifftwo}[3]{\ensuremath{\frac{\partial^2{#1}}{\partial{#2}
\, \partial{#3}}}}
\newcommand{\pdiff}[2]{\ensuremath{\frac{\partial^2{#1}}{\partial{#2}^2}}}
\newcommand{\pdifn}[3]{\ensuremath{\frac{\partial^{#1}{#2}}{\partial{#3}^{#1}}}}


\newcommand{\Fok}{\ensuremath{\mathcal{L}}}

\newcommand{\Ra}{\ensuremath{\mathit{Ra}}}


\newcommand{\unhyp}{\ensuremath{(0,1)^{\df(\meparam)}}}
\newcommand{\unhypthesis}{\ensuremath{(0,1)^{\df(\eparam)}}}

% Expected Value section
% use this command if no amsfonts
%\newcommand{\Exs}{\mathsf{I\!E}}
% use this command with amsfonts
\newcommand{\Exs}{\ensuremath{{\mathbb{E}}}}

% format shortcuts
\newcommand{\beq}{\begin{quotation}}
\newcommand{\enq}{\end{quotation}}

\newcommand{\mcite}[2]{\citetext{#1  \citealp{#2}}}
\newcommand{\incite}[2]{\citealp{#1,\citetext{#2}}}


\newcommand{\estart}{\begin{equation}}
\newcommand{\eend}{\end{equation}}

% example of new command with arguments
\newcommand{\pnorm}[2]{\|\, #1 \,\|_{#2}}
% to use type \pnorm{$x$}{2} for instance

% graphics commands: faster 
% #1=scaling ratio; #2 is file_name
%%\newcommand{\fastgraph}[2]{\includegraphics[keepaspectratio,scale=#1]{#2}}
\newcommand{\fastgraph}[2]{\includegraphics[scale=#1]{#2}}

\newcommand{\widgraph}[2]{\includegraphics[keepaspectratio,width=#1]{#2}}
%%\newcommand{\widgraph}[2]{\includegraphics[width=#1]{#2}}

\newcommand{\epsgraph}[2]{\epsfig{file=#2,width=#1}}

\newcommand{\scr}[1]{\ensuremath{\mathcal{#1}}}

\newcommand{\intzinf}{\int_{0}^{\infty}}
\newcommand{\edist}{\ensuremath{\overset{d}{=}}}
\newcommand{\erate}{\ensuremath{\overset{\cdot}{=}}}
\newcommand{\defntr}{\ensuremath{\triangleq}}
\newcommand{\defn}{\ensuremath{:  =}}
\newcommand{\rdefn}{\ensuremath{= :}}

\newcommand{\degree}{\ensuremath{d}}
\newcommand{\vertexsub}{\ensuremath{S}}

\newcommand{\ham}{\ensuremath{d_H}}
\newcommand{\app}{\ensuremath{\operatorname{APP}}}

\newcommand{\xvec}{\ensuremath{{{\mathbf{x}}}}}
\newcommand{\Xvec}{\ensuremath{{{\mathbf{X}}}}}
\newcommand{\qxvec}{\ensuremath{{{\mathbf{\estim{x}}}}}}
%\newcommand{\xvec}{x}
\newcommand{\bigX}{\ensuremath{{\mathbf{X}}}}
\newcommand{\bigY}{\ensuremath{{\mathbf{Y}}}}

\newcommand{\empav}[1]{\bar{#1}}

\newcommand{\xs}{\ensuremath{x^*}}

\newcommand{\xvecc}{\ensuremath{{\bf{x}}}}
%\newcommand{\xvecc}{\ensuremath{x}}

\newcommand{\gvec}{{\bf{u}}}
\newcommand{\mvec}{\ensuremath{Z}}
\newcommand{\yvec}{{\bf{y}}}
\newcommand{\uvec}{{\bf{u}}}
\newcommand{\zvec}{{\bf{z}}}
\newcommand{\wvec}{{\bf{w}}}
\newcommand{\vvec}{{\bf{v}}}

\newcommand{\Yvec}{{\bf{Y}}}
\newcommand{\Uvec}{{\bf{U}}}
\newcommand{\Zvec}{{\bf{Z}}}
\newcommand{\Wvec}{{\bf{W}}}
\newcommand{\Vvec}{{\bf{V}}}

\newcommand{\xsca}{{{x}}}
\newcommand{\ysca}{{{y}}}
\newcommand{\zsca}{{{z}}}
\newcommand{\wsca}{{{w}}}
\newcommand{\usca}{{{u}}}
\newcommand{\vsca}{{{v}}}

\newcommand{\Xsca}{{{X}}}
\newcommand{\Ysca}{{{Y}}}
\newcommand{\Zsca}{{{Z}}}
\newcommand{\Wsca}{{{W}}}
\newcommand{\Usca}{{{U}}}
\newcommand{\Vsca}{{{V}}}

\newcommand{\vect}[1]{{\bf{#1}}}


\newcommand{\ucovar}{\ensuremath{J}}

\newcommand{\gsca}{\ensuremath{u}}
\newcommand{\msca}{\ensuremath{z}}
\newcommand{\ssca}{\ensuremath{z}}


\newcommand{\psifam}{\ensuremath{\left\{2^m}
\psi^{m}_n(2^m t - n)\right\}}
\newcommand{\xcoeff}{\ensuremath{x^{m}_n}}
\newcommand{\psif}{\ensuremath{\psi(2^m t - n)}}

\newcommand{\updmn}[1]{\ensuremath{{#1}^{m}_{n}}}
\newcommand{\updn}[3]{\ensuremath{{#1}^{#2}_{#3}}}

\newcommand{\noise}{\ensuremath{w}}
\newcommand{\vnoise}{\ensuremath{W}}

\newcommand{\noiseb}{\ensuremath{\zeta}}
\newcommand{\noisec}{\ensuremath{v}}

\newcommand{\amat}{\ensuremath{A}}
\newcommand{\bmat}{\ensuremath{B}}
\newcommand{\cmat}{\ensuremath{C}}
\newcommand{\dmat}{\ensuremath{D}}

\newcommand{\wavdet}[2]{\ensuremath{x^{#1}_{#2}}}

% Short commands for centering; itemizing etc.

\newcommand{\bec}{\begin{center}}
\newcommand{\enc}{\end{center}}

\newcommand{\beit}{\begin{itemize}}
\newcommand{\enit}{\end{itemize}}

\newcommand{\been}{\begin{enumerate}}
\newcommand{\enen}{\end{enumerate}}


% Slide commands for seminar class
\newcommand{\comsl}{\begin{slide}}
\newcommand{\comspor}{\begin{slide*}}

\newcommand{\comsld}[2]{\begin{slide}[#1,#2]}
\newcommand{\comspord}[2]{\begin{slide*}[#1,#2]}

\newcommand{\mendsl}{\end{slide}}
\newcommand{\mendspo}{\end{slide*}}


\newcommand{\atgvec}{\ensuremath{\msca  \gvec}}
\newcommand{\atgsca}{\ensuremath{\msca  \gsca}}

\newcommand{\satgvec}{\ensuremath{\sqrt{\ssca}  \gvec}}
\newcommand{\satgsca}{\ensuremath{\sqrt{\ssca}  \gsca}}

\newcommand{\repvec}{\mbox{$\xvec \, \edist \, \atgvec$}}
\newcommand{\repsca}{\mbox{$\xsca \, \edist \, \atgsca$}}

\newcommand{\srepvec}{\mbox{$\xvec \, \edist \, \satgvec$}}
\newcommand{\srepsca}{\mbox{$\xsca \, \edist \, \satgsca$}}


\newcommand{\normal}[1]{\ensuremath{\mathcal{N}(0,#1)}}
\newcommand{\normall}[2]{\ensuremath{\mathcal{N}(#1,#2)}}

% Define estimates of quantities
\newcommand{\estim}[1]{\ensuremath{\widehat{#1}}}
\newcommand{\bls}[1]{\ensuremath{\widehat{#1}_{BLS}}}
\newcommand{\map}[1]{\ensuremath{\widehat{#1}_{MAP}}}
\newcommand{\wtil}[1]{\ensuremath{\widetilde{#1}}}

\newcommand{\mapfunc}{\ensuremath{f}}

\newcommand{\condv}{r}
\newcommand{\real}{\ensuremath{{\mathbb{R}}}}
\newcommand{\realp}{\ensuremath{{\mathbb{R^{+}}}}}
\newcommand{\posint}{\ensuremath{{\mathbb{N}}}}
\newcommand{\allint}{\ensuremath{{\mathbb{Z}}}}
% Define the (Gibbs) partition function
\newcommand{\parti}{{\mathbf Z}}

% Define integrals over various ranges
\newcommand{\intoi}{\ensuremath{\int_0^{\infty}}}
\newcommand{\intii}{\ensuremath{\int_{-\infty}^{\infty}}}
\newcommand{\intar}[2]{\ensuremath{\int_{#1}^{#2}}}

\newcommand{\lincon}{\ensuremath{{\mathcal{L}}}}

% Standard forms for certain densities
\newcommand{\student}[3]{\ensuremath{ (1+ \frac{#1^2}{2 #2^2})^{-#3}}}
\newcommand{\gamd}[1]{\ensuremath{Z({#1})}}
\newcommand{\symgam}[3]{\ensuremath{\left({#3}/({#3} + {#1}^2)\right)^{#2}}}
\newcommand{\gausss}[2]{\ensuremath{\frac{1}{\sqrt{2\pi} #2} \exp{(-\frac{#1^2}{2 #2^2})}}}
\newcommand{\gaussm}[3]{\ensuremath{\frac{1}{\sqrt{2\pi}
|#2|^{\frac{#3}{2}}} \exp{\big(-\frac{1}{2} #1'#2^{-1} #1 \big)}}}

%%% Partial derivatives
\newcommand{\dif}[2]{\ensuremath{\frac{\partial #1}{\partial #2}}}
\newcommand{\diff}[2]{\ensuremath{\frac{\partial^2 #1}{\partial #2^2}}}
\newcommand{\dift}[3]{\ensuremath{\frac{\partial^2 #1}{\partial #2 \partial #3}}}

%% Ordinaray derivatives
\newcommand{\odif}[2]{\ensuremath{\frac{d #1}{d #2}}}
\newcommand{\odiff}[2]{\ensuremath{\frac{d^2 #1}{d #2^2}}}
\newcommand{\odift}[3]{\ensuremath{\frac{d #1}{d #2 d #3}}}

\newcommand{\hself}{H-ss}
\newcommand{\asym}{\beta}
\newcommand{\pcon}[2]{p(#1 | #2)}
\newcommand{\scaleform}[3]{\frac{1}{#2} {#3}(\frac{#1}{#2})}


\newcommand{\scale}{\ensuremath{\lambda}}
\newcommand{\tail}{\ensuremath{\alpha}}

% Characteristic functions and Laplace transforms
\newcommand{\charfunc}[2]{\ensuremath{\phi_{#1}{(#2)}}}
% This second command is for putting the function w/o argument
\newcommand{\charf}[1]{\ensuremath{\phi_{#1}}}

\newcommand{\laptran}[2]{\ensuremath{\psi_{#1}{(#2)}}}

% Gaussian density function
\newcommand{\gaussd}[2]{\ensuremath{\frac{1}{\sqrt{2 \pi #1}}
\exp{(-\frac{#2^2}{2 #1})}}}





% Fiddly little text commands
\newcommand{\half}{\ensuremath{^{\frac{1}{2}}}}
\newcommand{\wien}{\ensuremath{w}}
\newcommand{\stat}{\ensuremath{x}}
\newcommand{\ustat}{\ensuremath{u}}

\newcommand{\ssp}{\ensuremath{\; \:}}

\newcommand{\iden}[1]{\ensuremath{I_{#1}}}
\newcommand{\zeros}[1]{\ensuremath{0_{#1}}}


% Funny accented words
\newcommand{\ito}{{I}t\^{o}\hspace*{3pt}}
\newcommand{\csis}{{C}sisz\'{a}r\hspace*{3pt}}
\newcommand{\csiss}{{C}sisz\'{a}r}




\DeclareMathOperator{\modd}{mod}
\DeclareMathOperator{\probab}{Pr}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\var}{var}
\DeclareMathOperator{\stable}{stable}
\DeclareMathOperator{\cov}{cov}
\DeclareMathOperator{\argmax}{argmax}
\DeclareMathOperator{\trace}{trace}
\DeclareMathOperator{\abs}{abs}
\DeclareMathOperator{\floor}{floor}
\DeclareMathOperator{\spanner}{span}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\child}{child}
\DeclareMathOperator{\parent}{parent}
\DeclareMathOperator{\tree}{tree}
\DeclareMathOperator{\loopy}{loop}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\for}{for}
\DeclareMathOperator{\all}{all}
\DeclareMathOperator{\some}{some}
\DeclareMathOperator{\cum}{cum}
\DeclareMathOperator{\toeplitz}{toeplitz}



